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IMAGE ENHANCEMENT IN SPATIAL DOMAIN

Spatial domain methods

Spatial domain refers to the image plane itself, and approaches in this category are based on

direct manipulation of pixels in an image. Frequency domain processing techniques are based on

modifying the Fourier transform of the image.

Suppose we have a digital image which can be represented by a two dimensional random field ),( yxf .

Spatial domain processes will be denoted by the expression

),(),( yxfTyxg or s=T(r)

The value of pixels, before and after processing, will be denoted by r and s, respectively.

Where f(x, y) is the input to the image, g(x, y) is the processed image and T is an operator on f.

The operator T applied on ),( yxf may be defined over:

(i) A single pixel ),( yx . In this case T is a grey level transformation (or mapping) function.

(ii) Some neighborhood of ),( yx .

(iii) T may operate to a set of input images instead of a single image.

Examples of Enhancement Techniques

Contrast Stretching

The result of the transformation shown in the figure below is to produce an image of higher contrast than

the original, by darkening the levels below m and brightening the levels above m in the original image.

This technique is known as contrast stretching.

Thresholding

The result of the transformation shown in the figure below is to produce a binary image.

One of the principal approaches is based on the use of so-called masks (also referred to as filters)

So, a mask/filter: is a small (say 3X3) 2-D array, such as the one shown in the figure, in which

the values of the mask coefficients determine the nature of the process, such as image

sharpening. Enhancement techniques based on this type of approach often are referred to as

mask processing or filtering.

Some Basic Grey Level Transform

We are dealing now with image processing methods that are based only on the intensity of single pixels.

Consider the following figure, which shows three basic types of functions used frequently for image

enhancement:

m

)(rTs

r

Fig: Some of the basic grey level transformation functions used for image enhancement

• The three basic types of functions used frequently for image enhancement:

1. Linear Functions:

• Negative Transformation

• Identity Transformation

2. Logarithmic Functions:

• Log Transformation

• Inverse-log Transformation

3. Power-Law Functions:

• nth power transformation

• nth root transformation

2.1.1 Identity Function

Output intensities are identical to input intensities

This function doesn’t have an effect on an image, it was included in the graph only for

completeness

Its expression:

s = r

2.1.2 Image Negatives

The negative of a digital image is obtained by the transformation function rLrTs 1)( shown in

the following figure, where L is the number of grey levels. The idea is that the intensity of the output

image decreases as the intensity of the input increases. This is useful in numerous applications such as

displaying medical images.

2.1.3 Log Transformation

The general form of the log transformation:

s = c log (1+r)

Where c is a constant, and r ≥ 0

Log curve maps a narrow range of low gray-level values in the input image into a wider

range of the output levels.

Used to expand the values of dark pixels in an image while compressing the higher-level

values.

It compresses the dynamic range of images with large variations in pixel values.

1L

1L

)(rTs

r

2.1.4 Inverse Logarithm Transformation

Do opposite to the log transformations

Used to expand the values of high pixels in an image while compressing the darker-level

values.

2.1.5 Power-law transformations

Power-law transformations have the basic form of:

s = c.rᵞ Where c and γ are positive constants

• Different transformation curves are obtained by varying γ (gamma)

Variety of devices used for image capture, printing and display respond according to a power

law. The process used to correct these power-law response phenomena is called gamma

correction.

For example, cathode ray tube (CRT) devices have an intensity-to-voltage response that is a

power function, with exponents varying from approximately 1.8 to 2.5.With reference to the

curve for γ=2.5 in Fig. 3.6, we see that such display systems would tend to produce images that

are darker than intended. This effect is illustrated in Fig. 3.7. Figure 3.7(a) shows a simple gray-

scale linear wedge input into a CRT monitor. As expected, the output of the monitor appears

darker than the input, as shown in Fig. 3.7(b). Gamma correction in this case is straightforward.

All we need to do is preprocess the input image before inputting it into the monitor by

performing the transformation. The result is shown in Fig. 3.7(c).When input into the same

monitor, this gamma-corrected input produces an output that is close in appearance to the

original image, as shown in Fig. 3.7(d).

In addition to gamma correction, power-law transformations are useful for general-purpose

contrast manipulation. See figure 3.8

Another illustration of Power-law transformation

Piecewise-Linear Transformation Functions

• Principle Advantage: Some important transformations can be formulated only as a

piecewise function.

• Principle Disadvantage: Their specification requires more user input that previous

transformations

• Types of Piecewise transformations are:

– Contrast Stretching

– Gray-level Slicing

– Bit-plane slicing

2.1.3 Contrast Stretching

Low contrast images occur often due to poor or non-uniform lighting conditions, or due to nonlinearity,

or small dynamic range of the imaging sensor. Contrast-stretching transformation, which is used to enhance the low contrast images

Figure 3.10(b) shows an 8-bit image with low contrast. Fig. 3.10(c) shows the result of contrast

stretching, obtained by setting (r1, s1) = (rmin, 0) and (r2, s2) = (rmax,L-1) where rmin and rmax

denote the minimum and maximum gray levels in the image, respectively. Thus, the

transformation function stretched the levels linearly from their original range to the full range [0,

L-1]. Finally, Fig. 3.10(d) shows the result of using the thresholding function defined previously,

with r1=r2=m, the mean gray level in the image.

• Figure 3.10(a) shows a typical transformation used for contrast stretching. The locations of points

(r1, s1) and (r2, s2) control the shape of the transformation function.

• If r1 = s1 and r2 = s2, the transformation is a linear function that produces no changes in gray

levels.

• If r1 = r2, s1 = 0 and s2 = L-1, the transformation becomes a thresholding function that

creates a binary image

• Intermediate values of (r1, s1) and (r2, s2) produce various degrees of spread in the gray levels of

the output image, thus affecting its contrast.

In general, r1 ≤ r2 and s1 ≤ s2 is assumed, so the function is always increasing

Gray-level Slicing

This technique is used to highlight a specific range of gray levels in a given image. It can be

implemented in several ways, but the two basic themes are:

One approach is to display a high value for all gray levels in the range of interest and a

low value for all other gray levels. This transformation, shown in Fig 3.11 (a), produces a

binary image.

The second approach, based on the transformation shown in Fig 3.11 (b), brightens the

desired range of gray levels but preserves gray levels unchanged.

Fig 3.11 (c) shows a gray scale image, and fig 3.11 (d) shows the result of using the

transformation in Fig 3.11 (a).

Bit-plane Slicing

Pixels are digital numbers, each one composed of 8 bits. Plane 0 contains the least significant bit

and plane 7 contains the most significant bit. Instead of highlighting gray-level range, we could

highlight the contribution made by each bit

• This method is useful and used in image compression.

Most significant bits contain the majority of visually significant data

2.2 Histogram processing. Definition of the histogram of an image.

By processing (modifying) the histogram of an image we can create a new image with specific

desired properties.

Suppose we have a digital image of size NN with grey levels in the range ]1,0[ L . The

histogram of the image is defined as the following discrete function:

n

nrp k

k )(

where

kr is the thk grey level, 1,,1,0 Lk

kn is the number of pixels in the image with grey level kr

n is the total number of pixels in the image

The histogram represents the frequency of occurrence of the various grey levels in the image. A

plot of this function for all values of k provides a global description of the appearance of the

image.

Fig: Four basic image types: (a) dark (b) light (c) low contrast (d) high contrast (e) Their

corresponding histograms

Images whose pixels occupy the entire range of intensity levels and tend to be distributed

uniformly will have an appearance of high contrast

It is possible to develop a transformation function that can automatically achieve this

effect, based on the histogram of the input image.

Types of processing:

Histogram equalization

Histogram matching (specification)

Local enhancement

Histogram equalisation (Automatic)

Consider transform of the form )(rTs 0≤ r ≤ 1

We assume that T(r) satisfies the following conditions

(a) T(r) monotonically increasing in range 0≤ r ≤ 1

(b) 0≤ T( r) ≤ 1 for 0≤ r ≤ 1

Fig: A grey level transformation function that is both single valued and

monotonically increasing.

The requirement in (a) that T(r) be single valued is needed to guarantee that the inverse

transformation will exist, and the monotonicity condition preserves the increasing order

from black to white in the output image

Condition (b) guarantees that the output gray levels will be in the same range as the input

levels.

The inverse transformation from s back to r is denoted )(1 sTr 0 ≤ S ≤ 1

---------------------------------------------------------------------------------------------------------------------------------

(Not in syllabus)

PDF (Probability of occurrence of a particular event) For example, what's the probability

for the temperature between 70and 80?

Definition of PDF

Let X be a continuous random variable. Then a probability distribution or probability

density function (pdf) of X is a function f (x) such that for any two numbers a and b with

a≤ b,

That is, the probability that X takes on a value in the interval [a; b] is the area above this

interval and under the graph of the density function. The graph of f (x) is often referred to

as the density curve.

------------------------------------------------------------------------------------------------------------

)(rpr denote the probability density function (pdf) of the variable r

)(sps denote the probability density function (pdf) of the variable s

If )(rpr and T(r) is known then )(sps can be obtained from

………… (a)

A transformation function of particular importance in image processing has the form

r

r dwwprTs0

)()( , 10 r (1)

By observing the transformation of equation (1) we immediately see that it possesses the following

properties:

(i) 10 s .

(ii) )()( 1212 rTrTrr , i.e., the function )(rT is increasing with r .

(iii) 0)()0(0

0

dwwpTs r and 1)()1(1

0

dwwpTs r . Moreover, if the original image has

intensities only within a certain range ] ,[ maxmin rr then 0)()(min

0min

r

r dwwprTs and

1)()(max

0max

r

r dwwprTs since maxmin and ,0)( rrrrrpr . Therefore, the new intensity s

takes always all values within the available range [0 1].

Suppose that )(rPr , )(sPs are the probability distribution functions (PDF’s) of the variables r and s

respectively.

(b)

From equation (a)

= =1 0≤ s ≤1

For discrete values we deal with probabilities and summations instead of probability density functions

and integrals. The probability of occurrence of gray level rk in an image is approximated by

2)(

N

nrp kk

k=0,1,2,….L-1

N is the total number of pixels in the image, nk is the number of pixels that have gray level rk and L is the

total number of possible gray level in the image.

Unfortunately, in a real life scenario we must deal with digital images. The discrete form of histogram

equalization is given by the relation

k

j

jk

j

jrn

nrPrTs

00

)()( Where k=0, 1,2,..L-1

Above equation is called histogram equalization or histogram linearization.

)()()(

0

rPdwwPdr

d

dr

rdT

dr

dsr

r

r

The inverse transform from s back to r is denoted by

k=0,1,2….L-1

Fig: (a) Images (b) Result of histogram equalization (c) Corresponding histogram.

Conclusion

From the above analysis it is obvious that the transformation of equation (1) converts the original image

into a new image with uniform probability density function. This means that in the new image all

intensities are present [look at property (iii) above] and with equal probabilities. The whole range of

intensities from the absolute black to the absolute white is explored and the new image will definitely

have higher contrast compared to the original image.

Histogram equalization may not always produce desirable results, particularly if the given histogram is

very narrow. It can produce false edges and regions. It can also increase image “graininess” and

“patchiness.”

)(1

kk sTr

Histogram Matching

Histogram equalization does not allow interactive image enhancement and generates only

one result: an approximation to a uniform histogram.

Suppose we want to specify a particular histogram shape (not necessarily uniform) which is

capable of highlighting certain grey levels in the image.

Let us suppose that Pr (r) and Pz (z) represent the PDFs of r and z where r is the input pixel

intensity level and z is the output pixel intensity

Suppose that histogram equalization is first applied on the original image r

k

j

jk

j

jrkkn

nrPrTs

00

)()( k=0,1,2…L-1

Where n is the total number of pixels in the image, nk is the number of pixels that have gray level rk and L

is the total number of possible gray level in the image.

Suppose that the desired image z is available and histogram equalization is applied as well

Finally

Or

Implementation

In order to see how histogram matching actually can be implemented, consider Fig. 3.19(a),

ignoring for a moment the connection shown between this figure and Fig. 3.19(c). Figure 3.19(a)

shows a hypothetical discrete transformation function s=T(r) obtained from a given image. The

first gray level in the image, r1, maps to s1; the second gray level, r2, maps to s2; the kth level

rk, maps to sk; and so on (the important point here is the ordered correspondence between these

values).

In order to implement histogram matching we have to go one step further. Figure 3.19(b) is a

hypothetical transformation function G obtained from a given histogram Pz(z) by using Eq.

(1)

For any zq, this transformation function yields a corresponding value vq. This mapping is shown

by the arrows in Fig. 3.19(b). Conversely, given any value vq, we would find the corresponding

value zq from G-1

. In terms of the figure, all this means graphically is that we would reverse the

direction of the arrows to map vq into its corresponding zq.

However, we know from the definition in Eq. (1) that v=s for corresponding subscripts, so we

can use exactly this process to find the zk corresponding to any value sk that we computed

previously from the equation r

r dwwprTs0

)()( . This idea is shown in Fig. 3.19(c).

Since we really do not have the z’s, we must resort to some sort of iterative scheme to find z

from s. From above equation we have G(zk)=sk, i.e G(zk)-sk=0,. Let

zzk for each value of k,

where is the smallest integer in the interval [0, L-1] such that 0)(

kszG where k=0,1…..L-1.

Minimum value of

z for which above equation is satisfied give zk.

The procedure we have just developed for histogram matching may be summarized as

follows:

1. Obtain the histogram equalization of the given image using equation

k

j

jk

j

jrkkn

nrPrTs

00

)()( to precompute a mapped level sk for each level rk

3. Obtain the transformation function G from the given pz(z) using

.

4. Precompute zk for each value of sk using the iterative scheme

5. For each pixel in the original image, if the value of that pixel is rk, map this value to its

corresponding level sk; then map level sk into the final level zk. Use the precomputed values from

Steps (2) and (4) for these mappings.

( Example)

Fig: (a) Image of the mars moon photo taken by NASA’s mars global surveyor (d)

Histogram.

Large concentration of pixels in the dark region of the histogram.

Fig: (a) Transformation function for histogram equalization (b) Histogram equalized image

(c) Histogram of (b)

Fig: (a) specified histogram (b) Curve (1) is from equation

k

j

jk

j

jrkkn

nrPrTs

00

)()( using the histogram in (a). curve (2) was obtained using

iterative procedure in (c) enhanced image using mapping from curve (2) (d) Histogram of

(c).

Local histogram equalisation

Global histogram equalisation is suitable for overall enhancement. It is often necessary to enhance details

over small areas.

The procedure is to define a square or rectangular neighbourhood and move the centre of this area from

pixel to pixel. At each location the histogram of the points in the neighbourhood is computed and a

histogram equalisation transformation function is obtained. This function is finally used to map the grey

level of the pixel centred in the neighbourhood. The centre of the neighbourhood region is then moved to

an adjacent pixel location and the procedure is repeated. Since only one new row or column of the

neighbourhood changes during a pixel-to-pixel translation of the region, updating the histogram obtained

in the previous location with the new data introduced at each motion step is possible quite easily. This

approach has obvious advantages over repeatedly computing the histogram over all pixels in the

neighbourhood region each time the region is moved one pixel location. Another approach often used to

reduce computation is to utilise non overlapping regions, but this methods usually produces an

undesirable checkerboard effect.

Let (x, y) be the coordinates of a pixel in an image, and let Sxy denote a neighborhood

(subimage) of specified size, centered at (x, y).The mean value of the pixels in Sxy can be

computed using the expression

where rs,t is the gray level at coordinates (s,t) in the neighborhood, and p(rs,t) is the neighborhood

normalized histogram component corresponding to that value of gray level. the gray-level

variance of the pixels in region Sxy is given by

The local mean is a measure of average grey level in neughborhood Sxy, and the varience (or

standard deviation) is a measure of contrast in that neighbourhood.

Figure 3.24 shows an SEM (scanning electron microscope) image of a tungsten filament

wrapped around a support. The filament in the center of the image and its support are quite clear

and easy to study. There is another filament structure on the right side of the image, but it is

much darker and its size and other features are not as easily discernable.

In this particular case, the problem is to enhance dark areas while leaving the light area as

unchanged as possible since it does not require enhancement. A measure of whether an area is

relatively light or dark at a point (x, y) is to compare the local average gray level xysm , to the

average image gray level, called the global mean and denoted MG. Thus, we have the first

element of our enhancement scheme: We will consider the pixel at a point (x, y) as a candidate

for processing if Gs Mkmxy 0 where k0 is a positive constant with value less than 1.0. Since we

are interested in enhancing areas that have low contrast. We also need a measure to determine

whether the contrast of an area makes it a candidate for enhancement. Thus, we will consider the

pixel at a point (x, y) as a candidate for enhancement if Gs Dkxy 2 where DG is the global

standard deviation and k2 is a positive constant. The value of this constant will be greater than

1.0 if we are interested in enhancing light areas and less than 1.0 for dark areas. Finally, we need

to restrict the lowest values of contrast we are willing to accept, otherwise the procedure would

attempt to enhance even constant areas, whose standard deviation is zero. Thus, we also set a

lower limit on the local standard deviation by requiring that xySGDk 1 with k1<k2. A pixel at

(x, y) that meets all the conditions for local enhancement is processed simply by multiplying it

by a specified constant, E, to increase (or decrease) the value of its gray level relative to the rest

of the image. The values of pixels that do not meet the enhancement conditions are left

unchanged. A summary of the enhancement method is as follows. Let f(x, y) represent the value

of an image pixel at any image coordinates (x, y), and let g(x, y) represent the corresponding

enhanced pixel at those coordinates.Then

),(

),(.),(

yxf

yxfEyxg if Gs Mkm

xy 0 and GSG DkDkxy 21

where, as indicated previously E, k0 , k1 , and k2 are specified parameters; MG is the global

mean of the input image; and DG is its global standard deviation.

Figure 3.25(a) shows the values ofxysm for all values of (x, y). Since the value of

xysm for each

(x, y) is the average of the neighboring pixels in a 3*3 area entered at (x, y), we expect the result

to be similar to the original image, but slightly blurred. Figure 3.25(b) shows in image formed

using all the values of xyS Similarly, we can construct an image out the values that multiply f(x,

y) at each coordinate pair (x, y) to form g(x, y). Since the values are either 1 or E, the image is

binary, as shown in Fig. 3.25(c). The dark areas correspond to 1 and the light areas to E.

Enhancement using Arithmetic/ Logic Operation

Arithmetic/logic operations involving images are performed on a pixel-by-pixel basis between

two or more images (this excludes the logic operation NOT, which is performed on a single

image). As an example, subtraction of two images results in a new image whose pixel at

coordinates (x, y) is the difference between the pixels in that same location in the two images

being subtracted. Depending on the hardware and/or software being used, the actual mechanics

of implementing arithmetic/logic operations can be done sequentially, one pixel at a time, or in

parallel, where all operations are performed simultaneously.

Logic operations similarly operate on a pixel-by-pixel basis. We need only be concerned with the

ability to implement the AND, OR, and NOT logic operators because these three operators are

functionally complete. In other words, any other logic operator can be implemented by using

only these three basic functions. When dealing with logic operations on gray-scale images, pixel

values are processed as strings of binary numbers. For example, performing the NOT operation

on a black, 8-bit pixel (a string of eight 0’s) produces a white pixel Intermediate values are

processed the same way, changing all 1’s to 0’s and vice versa. Thus, the NOT logic operator

performs the same function as the negative transformation. The AND and OR operations are

used for masking; that is, for selecting subimages in an image, as illustrated in Fig. 3.27. In the

AND and OR image masks, light represents a binary 1 and dark represents a binary 0. Masking

sometimes is referred to as region of interest(ROI) processing. In terms of enhancement,

masking is used primarily to isolate an area for processing. This is done to highlight that area and

differentiate it from the rest of the image.

Spatial domain: Enhancement in the case of many realisations of an image of

interest available

Image Subtraction

The difference between two image f(x,y) and h(x,y), expressed as

is obtained by computing the difference between all pairs of corresponding pixels from f and h.

The key usefulness of subtraction is the enhancement of differences between images. Figure

3.28(a) shows the fractal image used earlier to illustrate the concept of bit planes. Figure 3.28(b)

shows the result of discarding (setting to zero) the four least significant bit planes of the original

image. The images are nearly identical visually, with the exception of a very slight drop in

overall contrast due to less variability of the graylevel values in the image of Fig. 3.28(b). The

pixel-by-pixel difference between these two images is shown in Fig. 3.28(c). The differences in

pixel values are so small that the difference image appears nearly black when displayed on an 8-

bit display. In order to bring out more detail, we can perform a contrast stretching

transformation. We chose histogram equalization, but an appropriate power-law transformation

would have done the job also. The result is shown in Fig. 3.28(d).This is a very useful image for

evaluating the effect of setting to zero the lower-order planes.

Image averaging

Image averaging is obtained by finding the average of K images. It is applied in de-noising the

images.

Suppose that we have an image ),( yxf of size NM pixels corrupted by noise ),( yxn , so we obtain a

noisy image as follows.

),(),(),( yxnyxfyxg

Where noise has zero average value

If an image ),( yxg is formed by averaging K different noisy image

Then it follows that

and ),(2

),(2 1

yxyxg

K

Where is the expected value of and and 2

),( yx are the variance of and n,

all at coordinates (x, y). The standard deviation at any point in the average image is

),(),(

1yxyxg K

means that (output)approaches ) (input) as the number of

noisy images used in the averaging process increases.

An important application of image averaging is in the field of astronomy, where imaging with

very low light levels is routine, causing sensor noise frequently to render single images virtually

useless for analysis. Figure 3.30(a) shows an image of a galaxy pair called NGC 3314, taken by

NASA’s Hubble Space Telescope with a wide field planetary camera. . Figure 3.30(b) shows the

same image, but corrupted by uncorrelated Gaussian noise with zero mean and a standard

deviation of 64 gray levels. This image is useless for all practical purposes. Figures 3.30(c)

through (f) show the results of averaging 8, 16, 64, and 128 images, respectively. We see that the

result obtained with K=128 is reasonably close to the original in visual appearance